We must keep in mind that what we see Melchior and Lucca do is just the preliminary works though. Melchior says "Alright, let's get started" at the end of the scene, then the screen goes black and we jump directly to the moment when the repairing is finished. We don't see the actual repairing of the sword and how it is done. But of course, it's true that we don't see a smithy, a chimney, and those kind of things which should be needed, so... maybe.